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Chapter 4: Problem 31

Write each number in standard notation. $$1.15 \times 10^{-3}$$

Short Answer

Expert verified
The standard notation of \(1.15 \times 10^{-3}\) is \(0.00115\).

Step by step solution

01

Understand the scientific notation

The given number is in scientific notation. In a scientific notation, the number \(1.15\) is the coefficient and the \(10^{-3}\) represents the tenth power. When the power of 10 is negative, it means the decimal point in the coefficient has to be moved to the left.
02

Move the decimal point

The \(10^{-3}\) tells us to move the decimal point in the coefficient \(1.15\) three places to the left because the exponent in the power is negative. If there are not enough digits to the left of the decimal, as in this case, insert additional zeroes to fill the spaces.
03

Write the standard notation

After shifting the decimal point three places to the left, the coefficient becomes \(0.00115\). Thus, the standard notation for the given number \(1.15 \times 10^{-3}\) is \(0.00115\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Notation
Standard notation is the way of writing numbers in the most commonly understood form, without the use of exponents. When you convert from scientific notation to standard notation, you simplify the number for everyday use.
For instance, in the exercise where we have the number written as \(1.15 \times 10^{-3}\), we need to rewrite it in a more straightforward format. In standard notation, this means we will transform the scientific notation into a number that's easy to understand and interpret at a glance.
Through a short process, we achieve a more intuitive representation, staying true to the original value.
  • No need for exponents in standard notation.
  • The number is expressed with full decimal placement.
  • It becomes more intuitive for quick understanding.
Decimal Point
The decimal point is a critical part of understanding how numbers are represented both in scientific and standard notation. It's the tiny dot used to separate the whole number part from the fraction part.
In our exercise, the decimal point in the number \(1.15\) needs to be moved according to the exponent attached to the 10.
Moving the decimal point correctly is essential because it determines the real value of the said number. Moving it to the right or left changes the size of the conventional number.
  • A shift to the left decreases the number's value.
  • A shift to the right increases the number's value.
  • Number of spaces moved is determined by the exponent.
Exponent
In scientific notation, the exponent is key to understanding exactly how to adjust the decimal point in the coefficient. It indicates how many places the decimal point should move, and the direction depends on whether the exponent is positive or negative.
For the example in the exercise, \(10^{-3}\) tells us to move the decimal point three places to the left.
This implies the coefficient, \(1.15\), must shrink in size to form its standard notation of \(0.00115\). Understanding the role of the exponent is essential as it alters the scale of the number considerably.
  • Positive exponent means move decimal point to the right.
  • Negative exponent means move decimal point to the left.
  • Defines the scale of the number in scientific notation.

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